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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A000219 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A000219" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA000219">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A000219 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A000219 </div> <div class=seqname> Number of plane partitions (or planar partitions) of n. <br><font size=-1>(Formerly M2566 N1016)</font> </div> </div> <div class=scorerefs> 279 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, 8512309, 12733429, 18974973, 28175955, 41691046, 61484961, 90379784, 132441995, 193487501, 281846923</div> <div class=seqdatalinks> (<a href="/A000219/list">list</a>; <a href="/A000219/graph">graph</a>; <a href="/search?q=A000219+-id:A000219">refs</a>; <a href="/A000219/listen">listen</a>; <a href="/history?seq=A000219">history</a>; <a href="/search?q=id:A000219&fmt=text">text</a>; <a href="/A000219/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Two-dimensional partitions of n in which no row or column is longer than the one before it (compare <a href="/A001970" title="Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.">A001970</a>). E.g., a(4) = 13:</div> <div class=sectline>4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2</div> <div class=sectline>.....1....2.....1...1......1...11.1..1........ 11</div> <div class=sectline>....................1.............1..1</div> <div class=sectline>.....................................1</div> <div class=sectline>In the above, one also must require that rows & columns are nondecreasing, e.g., [1,1; 2] is also forbidden (which implies that row and column lengths are nondecreasing, if empty cells are identified with cells filled with 0's). - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 22 2018</div> <div class=sectline>Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner. - <a href="/wiki/User:Wouter_Meeussen">Wouter Meeussen</a></div> <div class=sectline>Also number of partitions of n objects of 2 colors, each part containing at least one black object; see example. - <a href="/wiki/User:Christian_G._Bower">Christian G. Bower</a>, Jan 08 2004</div> <div class=sectline>Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. E.g., n=3 gives 111, 12, 12', 3, 3', 3''. - <a href="/wiki/User:Jon_Perry">Jon Perry</a>, May 27 2004</div> <div class=sectline>The bijection between the partitions in the two preceding comments goes by identifying a part with k black objects with a part of type k. - <a href="/wiki/User:David_Scambler">David Scambler</a> and <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, May 01 2013</div> <div class=sectline>Can also be regarded as the number of Jordan canonical forms for an n X n matrix. (I.e., a 5 X 5 matrix has 24 distinct Jordan canonical forms, dependent on the algebraic and geometric multiplicity of each eigenvalue.) - Aaron Gable (agable(AT)hmc.edu), May 26 2009</div> <div class=sectline>(1/n) * convolution product of n terms * <a href="/A001157" title="a(n) = sigma_2(n): sum of squares of divisors of n.">A001157</a> (sum of squares of divisors of n): (1, 5, 10, 21, 26, 50, 50, 85, ...) = a(n). As shown by [Bressoud, p. 12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 48. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 13 2009</div> <div class=sectline>Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13, ...) = <a href="/A026007" title="Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n ar...">A026007</a>: (1, 1, 2, 5, 8, 16, 28, 49, 83, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jun 13 2009</div> <div class=sectline>Starting with offset 1 = row sums of triangle <a href="/A162453" title="Plane partition triangle, row sums = A000219; derived from the Euler transform of [1, 2, 3, ...].">A162453</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jul 03 2009</div> <div class=sectline>Unfortunately, Wright's formula is also incomplete in the paper by G. Almkvist: "Asymptotic formulas and generalized Dedekind sums", p. 344, (the denominator should have sqrt(3*Pi) not sqrt(Pi)). This error was already corrected in the paper by Steven Finch: "Integer Partitions". - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Aug 17 2015</div> <div class=sectline>Also the number of non-isomorphic weight-n chains of multisets whose dual is also a chain of multisets. The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. The weight of a multiset partition is the sum of sizes of its parts. - <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Sep 25 2018</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.</div> <div class=sectline>G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.</div> <div class=sectline>D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.</div> <div class=sectline>Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.</div> <div class=sectline>L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).</div> <div class=sectline>I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).</div> <div class=sectline>P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Royal Soc., 211 (1912), 345-373.</div> <div class=sectline>P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.</div> <div class=sectline>P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, May 21 2014</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Suresh Govindarajan, <a href="/A000219/b000219.txt">Table of n, a(n) for n = 0..6500</a> (first 401 terms from T. D. Noe)</div> <div class=sectline>G. Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.</div> <div class=sectline>G. E. Andrews and P. Paule, <a href="http://dx.doi.org/10.1112/jlms/jdm079">MacMahon's partition analysis XII: Plane Partitions</a>, J. Lond. Math. Soc., 76 (2007), 647-666.</div> <div class=sectline>A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="http://dx.doi.org/10.1017/S0305004100042171">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.</div> <div class=sectline>A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]</div> <div class=sectline>Michael Beeler, R. William Gosper and Richard C. Schroeppel, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18">HAKMEM, ITEM 18</a>, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.</div> <div class=sectline>Edward A. Bender, <a href="http://www.jstor.org/stable/2028691">Asymptotic methods in enumeration</a>, SIAM Review 16 (1974), no. 4, p. 509.</div> <div class=sectline>E. A. Bender and D. E. Knuth, <a href="http://dx.doi.org/10.1016/0097-3165(72)90007-6">Enumeration of Plane Partitions</a>, J. Combin. Theory A. 13, 40-54, 1972.</div> <div class=sectline>S. Benvenuti, B. Feng, A. Hanany and Y. H. He, <a href="http://arXiv.org/abs/hep-th/0608050">Counting BPS operators in gauge theories: Quivers, syzygies and plethystics</a>, arXiv:hep-th/0608050, p. 41-42.</div> <div class=sectline>Henry Bottomley, <a href="/A000219/a000219.gif">Illustration of initial terms</a></div> <div class=sectline>D. M. Bressoud and J. Propp, <a href="http://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.</div> <div class=sectline>Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, <a href="https://arxiv.org/abs/1606.07070">Twisted sectors from plane partitions</a>, arXiv preprint arXiv:1606.07070 [hep-th], 2016.</div> <div class=sectline>Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, <a href="https://arxiv.org/abs/2004.08901">Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions</a>, arXiv:2004.08901 [math.CO], 2020.</div> <div class=sectline>Steven Finch, <a href="/A000219/a000219_1.pdf">Integer Partitions</a>, September 22, 2004. [Cached copy, with permission of the author]</div> <div class=sectline>P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 580.</div> <div class=sectline>Bernhard Heim, Markus Neuhauser and Robert Tr枚ger, <a href="https://arxiv.org/abs/2109.15145">Inequalities for Plane Partitions</a>, arXiv:2109.15145 [math.CO], 2021.</div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=141">Encyclopedia of Combinatorial Structures 141</a></div> <div class=sectline>Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016, p. 18.</div> <div class=sectline>Vaclav Kotesovec, <a href="/A000219/a000219.jpg">Graphs - The asymptotic ratio (250000 terms)</a></div> <div class=sectline>D. E. Knuth, <a href="http://dx.doi.org/10.1090/S0025-5718-1970-0277401-7">A Note on Solid Partitions</a>, Math. Comp. 24, 955-961, 1970.</div> <div class=sectline>Oleg Lazarev, Matt Mizuhara and Ben Reid, <a href="http://www.math.oregonstate.edu/~swisherh/LazarevMizuharaReid.pdf">Some Results in Partitions, Plane Partitions, and Multipartitions</a>, 13 August 2010.</div> <div class=sectline>P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.</div> <div class=sectline>J. Mangual, <a href="http://arxiv.org/abs/1210.7109">McMahon's Formula via Free Fermions</a>, arXiv preprint arXiv:1210.7109 [math.CO], 2012. - From <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Jan 01 2013</div> <div class=sectline>Ville Mustonen and R. Rajesh, <a href="http://arXiv.org/abs/cond-mat/0303607">Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ...</a>, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.</div> <div class=sectline>L. Mutafchiev and E. Kamenov, <a href="https://arxiv.org/abs/math/0601253">On The Asymptotic Formula for the Number of Plane Partitions...</a>, arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.</div> <div class=sectline>Ken Ono, Sudhir Pujahari and Larry Rolen, <a href="https://arxiv.org/abs/2201.01352">Tur谩n inequalities for the plane partition function</a>, arXiv:2201.01352 [math.NT], 2022.</div> <div class=sectline>I. Pak, <a href="http://dx.doi.org/10.1007/s11139-006-9576-1">Partition bijections, a survey</a>, Ramanujan J. 12 (2006) 5-75.</div> <div class=sectline>A. Rovenchak, <a href="http://arxiv.org/abs/1401.4367">Enumeration of plane partitions with a restricted number of parts</a>, arXiv preprint arXiv:1401.4367 [math-ph], 2014.</div> <div class=sectline>Raphael Schumacher, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-2/Schumacher12132016.pdf">The self-counting identity</a>, Fib. Quart., 55 (No. 2 2017), 157-167.</div> <div class=sectline>N. J. A. Sloane, <a href="/transforms.txt">Transforms</a></div> <div class=sectline>J. Stienstra, <a href="https://arxiv.org/abs/math/0502197">Mahler measure, Eisenstein series and dimers</a>, arXiv:math/0502197 [math.NT], 2005.</div> <div class=sectline>Bal谩zs Szendr艖i, <a href="http://dx.doi.org/10.2140/gt.2008.12.1171">Non-commutative Donaldson-Thomas invariants and the conifold</a>, Geometry & Topology 12.2 (2008): 1171-1202.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a></div> <div class=sectline>E. M. Wright, <a href="https://doi.org/10.1112/jlms/s1-43.1.501">Rotatable partitions</a>, J. London Math. Soc., 43 (1968), 501-505.</div> <div class=sectline><a href="/index/Cor#core">Index entries for "core" sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.</div> <div class=sectline>Euler transform of sequence [1, 2, 3, ...].</div> <div class=sectline>a(n) ~ (c_2 / n^(25/36)) * exp( c_1 * n^(2/3) ), where c_1 = <a href="/A249387" title="Decimal expansion of the constant 'b' appearing in the asymptotic expression of the number of plane partitions of n as a*n^(...">A249387</a> = 2.00945... and c_2 = <a href="/A249386" title="Decimal expansion of the constant 'a' appearing in the asymptotic expression of the number of plane partitions of n as a*n^(...">A249386</a> = 0.23151... - Wright, 1931. Corrected Jun 01 2010 by Rod Canfield - see Mutafchiev and Kamenov. The exact value of c_2 is e^(2c)*2^(-11/36)*zeta(3)^(7/36)*(3*Pi)^(-1/2), where c = Integral_{y=0..inf} (y*log(y)/(e^(2*Pi*y)-1))dy = (1/2)*zeta'(-1).</div> <div class=sectline>The exact value of c_1 is 3*2^(-2/3)*Zeta(3)^(1/3) = 2.0094456608770137530649... - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Sep 14 2014</div> <div class=sectline>a(n) = (1/n) * Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n) = <a href="/A001157" title="a(n) = sigma_2(n): sum of squares of divisors of n.">A001157</a>(n) = sum of squares of divisors of n. - <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Jan 20 2002</div> <div class=sectline>G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Jan 10 2003</div> <div class=sectline>From <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Nov 07 2016: (Start)</div> <div class=sectline>More precise asymptotics: a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36) * n^(25/36))</div> <div class=sectline>* (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where</div> <div class=sectline>c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))</div> <div class=sectline>c2 = -0.02576771365117401620018082... = 353*Zeta(3)^(1/3)/(248832*2^(2/3)) - 17*Zeta(3)^(4/3)/(3225600*2^(2/3)) - 71575/(1492992*(2*Zeta(3))^(2/3))</div> <div class=sectline>c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000</div> <div class=sectline>and A = <a href="/A074962" title="Decimal expansion of Glaisher-Kinkelin constant A.">A074962</a> is the Glaisher-Kinkelin constant.</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>A planar partition of 13:</div> <div class=sectline> 4 3 1 1</div> <div class=sectline> 2 1</div> <div class=sectline> 1</div> <div class=sectline>a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+ 15*sigma_2(1)*sigma_2(2)^2+30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24<wbr>*sigma_2(5)) = 24. - <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Jan 10 2003</div> <div class=sectline>From <a href="/wiki/User:David_Scambler">David Scambler</a> and <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, May 01 2013: (Start)</div> <div class=sectline>There are a(4) = 13 partitions of 4 objects of 2 colors ('b' and 'w'), each part containing at least one black object:</div> <div class=sectline>1 black part:</div> <div class=sectline> [ bwww ]</div> <div class=sectline>2 black parts:</div> <div class=sectline> [ bbww ]</div> <div class=sectline> [ bww, b ]</div> <div class=sectline> [ bw, bw ]</div> <div class=sectline>3 black parts:</div> <div class=sectline> [ bbbw ]</div> <div class=sectline> [ bbw, b ]</div> <div class=sectline> [ bb, bw ]</div> <div class=sectline>(but not: [bw, bb ] )</div> <div class=sectline> [ bw, b, b ]</div> <div class=sectline>4 black parts:</div> <div class=sectline> [ bbbb ]</div> <div class=sectline> [ bbb, b ]</div> <div class=sectline> [ bb, bb ]</div> <div class=sectline> [ bb, b, b ]</div> <div class=sectline> [ b, b, b, b ]</div> <div class=sectline>(End)</div> <div class=sectline>From <a href="/wiki/User:Geoffrey_Critzer">Geoffrey Critzer</a>, Nov 29 2014: (Start)</div> <div class=sectline>The corresponding partitions of the integer 4 are:</div> <div class=sectline> 4'''</div> <div class=sectline> 4''</div> <div class=sectline> 3'' + 1</div> <div class=sectline> 2' + 2'</div> <div class=sectline> 4'</div> <div class=sectline> 3' + 1</div> <div class=sectline> 2 + 2'</div> <div class=sectline> 2' + 1 + 1</div> <div class=sectline> 4</div> <div class=sectline> 3 + 1</div> <div class=sectline> 2 + 2</div> <div class=sectline> 2 + 1 + 1</div> <div class=sectline> 1 + 1 + 1 + 1.</div> <div class=sectline>(End)</div> <div class=sectline>From <a href="/wiki/User:Gus_Wiseman">Gus Wiseman</a>, Sep 25 2018: (Start)</div> <div class=sectline>Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:</div> <div class=sectline> {{1,1,1,1}}</div> <div class=sectline> {{1,1,2,2}}</div> <div class=sectline> {{1,2,2,2}}</div> <div class=sectline> {{1,2,3,3}}</div> <div class=sectline> {{1,2,3,4}}</div> <div class=sectline> {{1},{1,1,1}}</div> <div class=sectline> {{2},{1,2,2}}</div> <div class=sectline> {{3},{1,2,3}}</div> <div class=sectline> {{1,1},{1,1}}</div> <div class=sectline> {{1,2},{1,2}}</div> <div class=sectline> {{1},{1},{1,1}}</div> <div class=sectline> {{2},{2},{1,2}}</div> <div class=sectline> {{1},{1},{1},{1}}</div> <div class=sectline>(End)</div> <div class=sectline>G.f. = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 48*x^6 + 86*x^7 + 160*x^8 + ...</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>series(mul((1-x^k)^(-k), k=1..64), x, 63);</div> <div class=sectline># second Maple program:</div> <div class=sectline>a:= proc(n) option remember; `if`(n=0, 1, add(</div> <div class=sectline> a(n-j)*numtheory[sigma][2](j), j=1..n)/n)</div> <div class=sectline> end:</div> <div class=sectline>seq(a(n), n=0..50); # <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Aug 17 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>CoefficientList[Series[Product[(1 - x^k)^-k, {k, 64}], {x, 0, 64}], x]</div> <div class=sectline>Zeta[3]^(7/36)/2^(11/36)/Sqrt[3 Pi]/Glaisher E^(3 Zeta[3]^(1/3) (n/2)^(2/3) + 1/12)/n^(25/36) (* asymptotic formula after Wright; <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Jun 23 2014 *)</div> <div class=sectline>a[0] = 1; a[n_] := a[n] = Sum[a[n - j] DivisorSigma[2, j], {j, n}]/n; Table[a[n], {n, 0, 50}] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Sep 21 2015, after <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a> *)</div> <div class=sectline>CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Feb 01 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, x^k / (1 - x^k)^2 / k, x * O(x^n))), n))}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jan 29 2005 */</div> <div class=sectline>(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-k), n))}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jan 29 2005 */</div> <div class=sectline>(PARI) my(N=66, x='x+O('x^N)); Vec( prod(n=1, N, (1-x^n)^-n) ) \\ <a href="/wiki/User:Joerg_Arndt">Joerg Arndt</a>, Mar 25 2014</div> <div class=sectline>(PARI) <a href="/A000219" title="Number of plane partitions (or planar partitions) of n.">A000219</a>(n)=#PlanePartitions(n) \\ See <a href="/A091298" title="Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.">A091298</a> for PlanePartitions(). For illustrative use: much slower than the above. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Sep 24 2018</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import cacheit</div> <div class=sectline>from sympy.ntheory import divisor_sigma</div> <div class=sectline>@cacheit</div> <div class=sectline>def <a href="/A000219" title="Number of plane partitions (or planar partitions) of n.">A000219</a>(n):</div> <div class=sectline> if n <= 1:</div> <div class=sectline> return 1</div> <div class=sectline> return sum(<a href="/A000219" title="Number of plane partitions (or planar partitions) of n.">A000219</a>(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n</div> <div class=sectline>print([<a href="/A000219" title="Number of plane partitions (or planar partitions) of n.">A000219</a>(n) for n in range(20)])</div> <div class=sectline># <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Oct 18 2009</div> <div class=sectline>(Julia)</div> <div class=sectline>using Nemo, Memoize</div> <div class=sectline>@memoize function a(n)</div> <div class=sectline> if n == 0 return 1 end</div> <div class=sectline> s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)</div> <div class=sectline> return div(s, n)</div> <div class=sectline>end</div> <div class=sectline>[a(n) for n in 0:20] # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, May 03 2020</div> <div class=sectline>(SageMath) # uses[EulerTransform from <a href="/A166861" title="Euler transform of Fibonacci numbers.">A166861</a>]</div> <div class=sectline>b = EulerTransform(lambda n: n)</div> <div class=sectline>print([b(n) for n in range(37)]) # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Nov 11 2020</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000784" title="Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one sy...">A000784</a>, <a href="/A000785" title="Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.">A000785</a>, <a href="/A000786" title="Number of inequivalent planar partitions of n, when considering them as 3D objects.">A000786</a>, <a href="/A005380" title="Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).">A005380</a>, <a href="/A005987" title="Number of symmetric plane partitions of n.">A005987</a>, <a href="/A048141" title="Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold...">A048141</a>, <a href="/A048142" title="Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a thre...">A048142</a>, <a href="/A089300" title="Number of planar partitions of n (A000219) that are non-squashing along rows and down columns (cf. A018819).">A089300</a>.</div> <div class=sectline>Cf. <a href="/A023871" title="Expansion of Product_{k>=1} (1 - x^k)^(-k^2).">A023871</a>-<a href="/A023878" title="Expansion of Product_{k>=1} (1 - x^k)^(-k^9).">A023878</a>, <a href="/A026007" title="Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n ar...">A026007</a>, <a href="/A001157" title="a(n) = sigma_2(n): sum of squares of divisors of n.">A001157</a>, <a href="/A162453" title="Plane partition triangle, row sums = A000219; derived from the Euler transform of [1, 2, 3, ...].">A162453</a>, <a href="/A285216" title="Indices of primes in A000219.">A285216</a>.</div> <div class=sectline>Differences: <a href="/A191659" title="First differences of A000219.">A191659</a>, <a href="/A191660" title="Second differences of A000219.">A191660</a>, <a href="/A191661" title="Third differences of A000219.">A191661</a>.</div> <div class=sectline>Row sums of <a href="/A089353" title="Triangle read by rows: T(n,m) = number of planar partitions of n with trace m.">A089353</a> and <a href="/A091438" title="Triangle a(n,k) of partitions of n objects of 2 colors, k of which are black and each part with at least one black object.">A091438</a> and <a href="/A091298" title="Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.">A091298</a>.</div> <div class=sectline>Column k=1 of <a href="/A144048" title="Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).">A144048</a>. - <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Nov 02 2012</div> <div class=sectline>Sequences "number of r-line partitions": <a href="/A000041" title="a(n) is the number of partitions of n (the partition numbers).">A000041</a> (r=1), <a href="/A000990" title="Number of plane partitions of n with at most two rows.">A000990</a> (r=2), <a href="/A000991" title="Number of 3-line partitions of n.">A000991</a> (r=3), <a href="/A002799" title="Number of 4-line partitions of n (i.e., planar partitions of n with at most 4 lines).">A002799</a> (r=4), <a href="/A001452" title="Number of 5-line partitions of n.">A001452</a> (r=5), <a href="/A225196" title="Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).">A225196</a> (r=6), <a href="/A225197" title="Number of 7-line partitions of n (i.e., planar partitions of n with at most 7 lines).">A225197</a> (r=7), <a href="/A225198" title="Number of 8-line partitions of n (i.e., planar partitions of n with at most 8 lines).">A225198</a> (r=8), <a href="/A225199" title="Number of 9-line partitions of n (i.e., planar partitions of n with at most 9 lines).">A225199</a> (r=9).</div> <div class=sectline>Cf. <a href="/A249386" title="Decimal expansion of the constant 'a' appearing in the asymptotic expression of the number of plane partitions of n as a*n^(...">A249386</a>, <a href="/A249387" title="Decimal expansion of the constant 'b' appearing in the asymptotic expression of the number of plane partitions of n as a*n^(...">A249387</a>.</div> <div class=sectline>Cf. <a href="/A161870" title="Convolution square of A000219.">A161870</a>, <a href="/A255610" title="G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).">A255610</a>, <a href="/A255611" title="G.f.: Product_{k>=1} 1/(1-x^k)^(4*k).">A255611</a>, <a href="/A255612" title="G.f.: Product_{k>=1} 1/(1-x^k)^(5*k).">A255612</a>, <a href="/A255613" title="G.f.: Product_{k>=1} 1/(1-x^k)^(6*k).">A255613</a>, <a href="/A255614" title="G.f.: Product_{k>=1} 1/(1-x^k)^(7*k).">A255614</a>, <a href="/A193427" title="G.f.: Product_{k>=1} 1/(1-x^k)^(8*k).">A193427</a>.</div> <div class=sectline>Sequence in context: <a href="/A225197" title="Number of 7-line partitions of n (i.e., planar partitions of n with at most 7 lines).">A225197</a> <a href="/A225198" title="Number of 8-line partitions of n (i.e., planar partitions of n with at most 8 lines).">A225198</a> <a href="/A225199" title="Number of 9-line partitions of n (i.e., planar partitions of n with at most 9 lines).">A225199</a> * <a href="/A356941" title="Number of multiset partitions of integer partitions of n such that all blocks are gapless.">A356941</a> <a href="/A191782" title="Sum of the lengths of the first ascents in all n-length left factors of Dyck paths.">A191782</a> <a href="/A358905" title="Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.">A358905</a></div> <div class=sectline>Adjacent sequences: <a href="/A000216" title="Take sum of squares of digits of previous term, starting with 2.">A000216</a> <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a> <a href="/A000218" title="Take sum of squares of digits of previous term; start with 3.">A000218</a> * <a href="/A000220" title="Number of asymmetric trees with n nodes (also called identity trees).">A000220</a> <a href="/A000221" title="Take sum of squares of digits of previous term; start with 5.">A000221</a> <a href="/A000222" title="Coefficients of m茅nage hit polynomials.">A000222</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="an important sequence">core</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Corrected by <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Jul 29 2006</div> <div class=sectline>Minor edits by <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Oct 27 2014</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified March 13 17:52 EDT 2025. Contains 381739 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>